The closure in Lip \(\alpha\) norms of rational modules with three generators (Q1059819)

If \(g_ 1,g_ 2,...,g_ n\) are functions on a compact subset X of the complex plane, we denote by \(R^{\alpha}(X,g_ 1,g_ 2,...,g_ n)\) the closure in the Lip(\(\alpha)\) norm of the set of functions \(h_ 0+h_ 1g_ 1+...+h_ ng_ n\), where \(h_ i\) run over rational functions without poles in X. In a preceding work, \textit{J. L.-M. Wang} proved that there is a compact X, \(\overset\circ X = \emptyset\), such that \(R^{\alpha}(X,\bar z)\neq lip(\alpha,X),\) \(0<\alpha <1\) [Ill. J. Math. 26, 632-636 (1982; Zbl 0487.41024)]. In this paper we deal with spaces of type \(R^{\alpha}(X,g_ 1,g_ 2).\) The main result is a characterization of \(R^{\alpha}(X,g,F\circ g),\) where F is a holomorphic function such that F''\(\neq 0\) and \(\overset\circ X = \emptyset\), in terms of the compact set \(Z=\{w\in X/{\bar \partial}g(w)=0\}\) (Theorem 1). Also we characterize when the space \(R^{\alpha}(X,\bar h,F\circ h)\) is dense in lip(\(\alpha\),X), F being of class \(C^ 2\) and h an analytic function (Theorem 2). Proofs of these theorems are by duality and the main tool is the construction of adequate kernels.